Question: A right pyramid has a square base with side length 10 cm. Its peak is 12 cm above the center of its base. What is the total surface area of the pyramid, in square centimeters?
Answer: [asy]
import three;
triple A = (0,0,0);
triple B = (1,0,0);
triple C = (1,1,0);
triple D = (0,1,0);
triple P = (0.5,0.5,1);
draw(B--C--D--P--B);
draw(P--C);
draw(B--A--D,dashed);
draw(P--A,dashed);
label("$A$",A,NW);
label("$B$",B,W);
label("$C$",C,S);
label("$D$",D,E);
label("$P$",P,N);
triple F= (0.5,0.5,0);
triple M=(B+C)/2;
draw(P--F--M,dashed);
draw(P--M);
label("$F$",F,S);
label("$M$",M,SW);
[/asy]

Let $F$ be the center of the square base and $M$ be the midpoint of an edge of the square, as shown.  Since the pyramid is a right pyramid, triangle $PFM$ is a right triangle.  We are given $PF = 12$, and we have $FM = (1/2)(AB) = 5$, so the Pythagorean Theorem gives us $PM = \sqrt{PF^2 + FM^2} = 13$.  Therefore, since the four lateral faces are congruent triangles, the total surface area of the pyramid is \[[ABCD] + 4[PAB] = 10^2 + 4(13)(10)/2 = \boxed{360}\text{ square centimeters}.\]